HINT: <no title>
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Notice there is a right angle in this diagram at chord RG⎯⎯⎯⎯⎯⎯⎯⎯. This means you can use the perpendicular bisector theorem.
STEP: Use the perpendicular bisector theorem
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Let's organise the information before we get started. We know:
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the length of RG⎯⎯⎯⎯⎯⎯⎯⎯=8
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length of OQ⎯⎯⎯⎯⎯⎯⎯⎯⎯=3
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OH⎯⎯⎯⎯⎯⎯⎯⎯⎯ is perpendicular to RG⎯⎯⎯⎯⎯⎯⎯⎯
and we want:
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the length of the side OH⎯⎯⎯⎯⎯⎯⎯⎯⎯ which is the radius of the circle
There are several ways to tackle this problem! Let's
look in the figure to see if there is a theorem about circles which can
be helpful.
We have the length of the chord RG⎯⎯⎯⎯⎯⎯⎯⎯, and there is a line from the centre of the circle O perpendicular to the chord, meeting the chord at point Q.
What a surprise... no, not really: we can use the theorem about the
perpendicular bisectors through the centre of a circle. In this case,
it tells us that RQ⎯⎯⎯⎯⎯⎯⎯⎯ and QG⎯⎯⎯⎯⎯⎯⎯⎯⎯ are half the size of RG⎯⎯⎯⎯⎯⎯⎯⎯, which is 8.
∴RQ⎯⎯⎯⎯⎯⎯⎯⎯=QG⎯⎯⎯⎯⎯⎯⎯⎯⎯=RG⎯⎯⎯⎯⎯⎯⎯⎯2(⊥line from centre bisects chord)=(8)2=4
STEP: Set up the information you have about a right-angled triangle
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It is also important to notice that OH⎯⎯⎯⎯⎯⎯⎯⎯⎯ is the radius of the circle; and more importantly, it is equal to any other radius of the circle, e.g. OH⎯⎯⎯⎯⎯⎯⎯⎯⎯≡OR⎯⎯⎯⎯⎯⎯⎯⎯. This is also the line segment you want!.
The key to opening this solution is to see that
there is a right-angled triangle in the figure. Can you see it? Line
segment OQ⎯⎯⎯⎯⎯⎯⎯⎯⎯ is one of the sides of this triangle.
Now we have three sides of a right-angled triangle.
STEP: Use the Theorem of Pythagoras
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We want one of the sides of a right triangle: it is
Pythagoras time! (We will not get the answer using sine, cosine or
tangent because we don't know either of the acute angles in the
triangle.)
c2OR⎯⎯⎯⎯⎯⎯⎯⎯2OR⎯⎯⎯⎯⎯⎯⎯⎯2∴OR⎯⎯⎯⎯⎯⎯⎯⎯=a2+b2=OQ⎯⎯⎯⎯⎯⎯⎯⎯⎯2+QR⎯⎯⎯⎯⎯⎯⎯⎯2=(3)2+(4)2=±(9)+(16)‾‾‾‾‾‾‾‾‾√=±25‾‾‾√=±5
Cool! We can throw out the negative sign, because this number is a distance - it has to be positive!
But wait! The question asked for the length of OH⎯⎯⎯⎯⎯⎯⎯⎯⎯ and we just found the length of OR⎯⎯⎯⎯⎯⎯⎯⎯. Was this a waste of time? No! You have found the radius of the circle: OR⎯⎯⎯⎯⎯⎯⎯⎯=25‾‾‾√; and OH⎯⎯⎯⎯⎯⎯⎯⎯⎯ is also a radius of the circle, so it has the same length! Therefore, OH⎯⎯⎯⎯⎯⎯⎯⎯⎯=25‾‾‾√.
The question says that you should round the answer
to two decimal places; but in this case the answer works out to be an
integer, so there is no surd or decimals to deal with.
The final answer is: OH⎯⎯⎯⎯⎯⎯⎯⎯⎯=5.
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